Maps on spaces of symmetric matrices preserving idempotence

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let Mn(F) be the space of all n×n matrices over F, and let Sn(F) be the subset of Mn(F) consisting of all symmetric matrices. Let V∈{Sn(F),Mn(F)}, a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,B∈V and λ∈F. It is shown that: when the characteristic of F is not 2, |F|>3 and n⩾4, Φ:Sn(F)→Sn(F) is a map preserving idempotence if and only if there exists an invertible matrix P∈Mn(F) such that Φ(A)=PAP-1 for every A∈Sn(F) and PtP=aIn for some nonzero scalar a in F.